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多线性形式最大值的尾概率及其应用AI 解读

Let Z be a k­way array consisting of independent standard normal variables. For column vectors h1, …, hk, define a multilinear form of degree k by $(h_1 \otimes \cdots \otimes h_k)' \vec(Z)$. We derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the hi’s under the condition that the hi’s are unit vectors, and of its standardized statistic obtained by dividing by the norm of Z. We also give formulas for the maximum of a symmetric…

光滑或分片光滑锥备择下卡方混合分布的权重AI 解读

We study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives. We show that the mixture weights of the $\bar{\chi}^2$ distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual. The weights can be calculated by integration involving the second fundamental form on the boundary of the cone. We illustrate our technique by examples involving a…